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UNIVERSITÀ DEGLI STUDI Quantum scattering calculated easily DI TRENTO G.P. Karwasz 1,2 H. Nowakowska3 1Dipartimento di Fisica, Università di Trento, 38050 Povo, Italy and Pomeranian Pedagogical Academy, 76-200 Slupsk, Poland now at: Institut für Chemie-Physikalische und Theoretische Chemie, Freie Universität Berlin, 14195 Berlin 3 The Szewalski Institute of Fluid-Flow Machinery Polish Academy of Science, 80-952 Gdansk, Poland Electrons and waves 1. Wave phenomena are characterized by interference. The impression of an acute dissonance happens when beats are below 50 Hz in frequency. 2. Quantum mechanics is another example of wave interference. In a scattering processes, the monochromatic, well-collimated beam of particles corresponds to a plane de Broglie wave Ψ0 = exp(ikz), with k being the wave number. 3. Following Huyghens’ principle, the scattering center acts as a source of spherical wave Ψ’ = exp(ikr). 4. Obviously, the scattered wave need not be perfectly spherical, so we add an angular factor Ψ’ = f (θ) exp(ikr). Angular distribution of scattered electrons are complicated functions of energy. Ramsauer’s effect Carl Ramsauer, in Gdansk, 1921 was the first who showed that electrons behave like waves: at some energies the gases like Ar, Kr become transparent to them: the cross section shows a minimum. This is a wave-like effect. In 1931, in Berlin- Reinickendorf, C. Ramsauer and R. Kollath measured angular distribution of scattered electrons, confirming Quantum wave mechanics. Coleman (1980) Sinapius (1980) Kauppila (1976) Canter (1974) Charlton (1984) Stein (1992) Present 2 Total cross section (10 m ) Positron scattering FIRST ACCELERATOR -20 Positrons are electrons with a positive charge. A long discussion lasted, if positrons also show Ramsauer minimum. New measurements from Trento showed surpsisingly that the (integral) cross section reamains constant, like predicted by Classical Mechanics for hard sphere scattering. REMODERATOR STAGE 10 9 8 7 Nakanishi (1986) Gianturco (1993) McEachran (1979) 6 5 4 3 Argon (Ar) INJECTION OPTICS 2 1 DEFLECTOR 10 Positron energy (eV) The scattering amplitude is given by: where is the angular momentum in the collision, =0,1,2…, are the ”phase shifts” of the respective partial waves and P(cos ) are Legendre polynomials The integral cross-section is given by 4 int 2 2 1sin 2 k 0 Quantum or classical? A “small” problem arises: if scattering is classical, angular distributions should be uniform in angle. This is not the case! To calculate angular distributions, the plane wave is developed into spherical, partial waves and a change in the phase of the scattered wave due to the potential is obtained from Schrödinger’s equation. But the scattering potential is unknown… 0.5 2 0 -20 2 1exp 2i 1P cos Differential cross section (10 m ) 1 f 2ik 5eV 8.7 eV 15 eV 0.4 0.3 0.2 0.1 0.0 0 A module in EXCELL has been constructed allowing to adjust phase shifts. Now, we can reproduce any angular distribution with a few partial waves! 20 40 60 80 100 120 140 160 180 Scattering angle (deg) The model shows that angular distributions can be reproduced, and integral ones also, but phase shifts are somewhat artificial. An inelastic process must be added! Why Quantum Mechanics adjusts itself to a classical result?